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## AP®︎ Calculus AB (2017 edition)

### Course: AP®︎ Calculus AB (2017 edition)>Unit 2

Lesson 4: Formal definition of the derivative as a limit

# The derivative of x² at any point using the formal definition

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.B (LO)
,
CHA‑2.B.2 (EK)
,
CHA‑2.B.3 (EK)
,
CHA‑2.B.4 (EK)
Sal finds the limit expression for the derivative of f(x)=x² at any point x and simplifies it into a simpler expression (spoiler: it's 2x). Created by Sal Khan.

## Want to join the conversation?

• Why does he call the lines Tangent and Secant? What is the relation to Trig?
• Tangent is the line that 'touchs' the curve
Secant is the line that 'cuts' the curve
The curve is also sometimes called 'sine' (From latin 'sinus' meaning curve)

Full explanation of relation to trig: http://mathforum.org/library/drmath/view/54053.html
• could you ever have a Derivative being "no slope" I know that a slope of zero is the X axis, but isn't a line of no slope the Y axis? Is my question of merit? or is it nonsesne?
• When a line or curve has a slope parallel to the y-axis, the slope is undefined, because you are dividing by zero. If you have a curve with a vertical tangent line at one point, you will see the derivative approach infinity for that point.
• 'd' doesn't stand for delta, does it?
That would be too simple
• The 'd' in fact does come from delta(Δ). dy/dx is similar to Δy/Δx which is the slope formula (that is rise over run).
Δy/Δx =(y2-y1)/(x2-x1) but in dy/dx the difference between the two points like x2 and x1 is taken to be much smaller or more accurately using limits to approach 0 for getting the slope at a single point.
The derivative so gives the slope of a tangent line that touches a curve only once. The slope formula in contrast gives the slope of a secant ( a line that intersects a curve twice) through a curve as it uses two points with significant difference.
The small delta symbol (δ) is also sometimes used to show a very very small change.
• At , why does h disappear because one takes the limit as h approaches 0?
• That is a result of the Direct Substitution Property which says "If f is a polynomial or a rational function and a is in the domain of f, then lim x->a f(x) = f(a)"

In this case, x is replaced by h and a is zero, so h disappears because we substitute 0 for h. That is why it was necessary to "clear" h from the denominator or the difference quotient. So that when we do direct substitution, we do not end up dividing by zero, which is undefined.
• Where does the word 'calculus' come from?
• It's true; the word calculus means pebble in Latin. In ancient Rome, pebbles were used to perform arithmetic (they didn't have calculus or even algebra), so this also gives us the word "calculate." Doing any math become closely associated with pebbles, and as a result, we got the word calculate and we began calling a specific type of math calculus even though calculus as we know it today never had anything to do with pebbles.
• I am wondering if derivative is a slope of some function, couldn't we put that slope in a general function formula to form a function which derivative we took? Something like this f(x)=f '(x) * x + b
But then 2x*x should equal x^2, but it does not. Where am I wrong?
• And now this is very useful. I just realized these two topics aren't in the videos so I suggested it.
1. THis is useful for something we called linear approximation. We know the tangent lines so we can now approximate some close x values. This is very useful when you don't have a calculator and also for newton's method which is point 2.
f(x) ~ f(a) + f'(a) (x-a)
a would be a nice close number.
2. It's used for newton's method.
http://en.wikipedia.org/wiki/Newton's_method
This is a method used for finding zeroes. Sometimes zeroes are very hard to find without a graphing calculator. You just can't use algebra to find it. SO this approximation is a algorithm thing. SO you keep doing again and again and usually you get closer to the exact number.
• at , how does (x+h)^2 become (x^2) + 2xh + (h^2) ?
• (x+h)^2 = (x+h) * (x+h) Multiplying these together you get (x^2)+2xh +(h^2)
• At , we simplify
lim h->0 [ (2xh + h^2) / h ]
by dividing the numerator by the denominator, h, to leave
lim h->0 [ 2x + h ].
In some of the previous limit videos, when we had a variable in the denominator that approached zero, we had to specify that the limit would be true so long as the variable was not ACTUALLY equal to zero. Why do we not specify that h is not equal to zero in this case?

• We do not actually SET h = 0, we just make h TEND to 0 so divisibility by h is acceptable.